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Variational integrators in discrete vakonomic mechanics

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Abstract

We introduce the discrete counterpart of the vakonomic method in Lagrangian mechanics with non-holonomic constraints. After defining the concepts of “admissible section” and “admissible infinitesimal variation” of a discrete vakonomic system, we aim to determinate those admissible sections that are critical for the Lagrangian of the system with respect to admissible infinitesimal variations. For sections that satisfy a certain regularity condition, we prove that critical sections are extremals of a variational problem without constraints canonically associated to the initial system (Lagrange multiplier rule). We introduce a notion of “constrained variational integrator”, which is characterized by a Cartan equation that ensures its simplecticity. Moreover, under certain regularity conditions we prove that these integrators can be locally constructed from a generating function of the second kind in the sense of symplectic geometry. Finally, the theory is illustrated with two elementary examples: an isoperimetric problem and an optimal control problem.

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Authors and Affiliations

  1. Department of Mathematics, University of Salamanca, 37008, Salamanca, Spain

    Pedro L. García

  2. Department of Applied Mathematics, University of Salamanca, 37008, Salamanca, Spain

    Antonio Fernández

  3. CINAMIL, Academia Militar, 2720-113, Amadora, Portugal

    César Rodrigo

Authors
  1. Pedro L. García
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  2. Antonio Fernández
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  3. César Rodrigo
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Correspondence to Pedro L. García.

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García, P.L., Fernández, A. & Rodrigo, C. Variational integrators in discrete vakonomic mechanics. RACSAM 106, 137–159 (2012). https://doi.org/10.1007/s13398-011-0030-x

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  • DOI: https://doi.org/10.1007/s13398-011-0030-x

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